Assign material to part of mesh
You only have one material assigned to your object.  To assign multiple materials to different parts of a mesh you must have multiple materials assigned to the object.

At the top of the Material properties tab there is a selection box which shows all materials currently available to the object.  This does not mean that all these materials are used on the object, it means they are available to be assigned to parts of the object.  By default, the entire mesh uses the first material in this list.



On your object, there was only one material ("Transparent") in this list.  To add a material to this list, click the + button to the right of the material, you can then select a portion of the mesh and hit the Assign button to assign the selected material to that portion of the mesh.

Under the Settings rollout there are options to change the material's color in the viewport.  It is often useful to change this for some material[s] when there are multiple materials on an object so you can see which ones are on which parts.